Here is the full list of all 42 Puyo Puyo Tetris achievements.
Popped 100,000
Popped 50,000
Performed an All Clear 150 times in completed matches
Performed an All Clear 100 times in completed matches
Performed a 7-Chain 150 times in completed matches
Cleared 100,000
Performed a 6-Chain 100 times in completed matches
Popped 10,000
100% adventure mode
Performed an All Clear 50 times in completed matches
Complete Adventure
70% adventure mode
Performed a 5-Chain 100 times in completed matches
50% adventure mode
Won ten Puzzle League matches
Cleared 50,000
Won your first Free Play match
Performed a 4-Chain 100 times in completed matches
perfect clear 3 times
Performed a 3-Chain 100 times in completed matches
Won your first Puzzle League match
Played all 3 Puyo Puyo Challenge modes
Played all 5 Arcade modes
Performed a Back-to-back 100 times in competed matches
Clear 200
Defeated 15 opponents in an Endurance match
Cleared 4 different-colored
Performed a Back-to-back 50 times in competed matches
perfect clear
8-Combo
Played all 3 Tetris Challenge modes
Cleared 10,000
Clear 100
Defeated 3 opponents in an Endurance match
Defeated 5 opponents in an Endurance match
Defeated 10 opponents in an Endurance match
Popped 1,000
Cleared 3 different-colored
Clear 50
Performed a Back-to-back 25 times in competed matches
5-Combo
Cleared 1,000